I think folks are trying to read too much razzle dazzle into CL (or, for that matter, any non-dimensional quantity used in engineering/physics/etc).
Suggest that a brief read on the subject of 'Dimensional Analysis' might be useful. See, for example,
DA_unified.pdf (mit.edu)
There is a number of important non-dimensional quantities in engineering (such as mach number, reynolds number, lift coefficient, drag coefficient, and so it goes on ..). In the real olden days, before this all came about, we had reams and reams and reams and yet more reams of experimental data from which to figure out stuff with our pencils and paper. Eventually, brighter minds prevailed and dimensionless quantities came to be. The main value to us is that, by using these animals, we can reduce the amount of experimentally reported data by orders of magnitude and that was a great development.
There is nothing terribly magical about CL; as Megan noted, it is just a ratio of measured lift (usually what we get from wind tunnel work) to some other useful quantities which make for a non-dimensional result. Of sideline note, the 'S' for area is just some representative area - it doesn't matter all that much what area is used as it only needs to be there to complete the non-dimensionalising process. Obviously, it makes logical sense to use some sort of rationally pertinent area - while we could easily use the area of the local football field, wing planform area makes for a sensible quantity. It follows that any sets of data for CL should specify what area is being used as it will change the quantum of the CL numerical values which result from the exercise. Similarly, we might use frontal area for CD numbers .... same, same .... but we really need to specify what area it is that we are using lest our paper and pencil work comes up with strange numbers.
As others have stated or inferred, the CL with which we normally work only applies to the typical low speed aircraft. If you read up on the background, CL also is tied up with Mach Number and Reynolds Number which need to be incorporated when relevant. A simple story is given at
The Lift Coefficient (nasa.gov)